0

Let $X$ be an affine variety. Its coordinate ring $k[X]$ is a quotient of some overall polynomial ring $R$. With $R$, we can define algebraic varieties, the Zariski topology, etc. How much of that can we still do if we start with $k[X]$?

For example, we can look at the ideals of $k[X]$ and see their vanishing locii as subsets of $X$. Do we still have:

  • weak nullstellensatz?
  • a correspondence between algebraic sets and radical ideals, between varieties and prime ideals, etc.?
  • a Noetherian topology/Noetherian domain?

It would be a tedious (though maybe informative) exercise to go through all of the results in my course notes and check that they still hold for $k[X]$. But is there a simpler way? Am I missing some obvious way to do this, or some place where this fails?

Sam Jaques
  • 2,090
  • All of the things you ask should be very straightforward (either from the definitions or from the correspondence theorem for ideals). What have you tried? – KReiser Jun 30 '20 at 10:48
  • I showed that ideals in $k[X]$ are of the form $J+I(X)$ for ideals $J$ in the overall ring, which gives a lot of the results fairly easily. But this seems rather ad hoc. Is there not some deeper correspondence? – Sam Jaques Jun 30 '20 at 11:18
  • Are you sure you're looking at the right correspondence theorem? This should make things clearer. – KReiser Jun 30 '20 at 18:24

0 Answers0